To find the value of \(k\) for which the equation \(gf(x) = 0\) has two equal roots, we need to substitute \(f(x)\) into \(g(x)\).

\(gf(x) = g(f(x)) = 2(2x^2 - 12x + 7) + k = 0\)

This simplifies to:

\(4x^2 - 24x + 14 + k = 0\)

For the quadratic equation \(ax^2 + bx + c = 0\) to have two equal roots, the discriminant must be zero:

\(b^2 - 4ac = 0\)

Here, \(a = 4\), \(b = -24\), and \(c = 14 + k\).

Substitute these values into the discriminant formula:

\((-24)^2 - 4 \times 4 \times (14 + k) = 0\)

\(576 - 16(14 + k) = 0\)

\(576 - 224 - 16k = 0\)

\(352 = 16k\)

\(k = \frac{352}{16} = 22\)